Instability of standing waves for a quasi-linear Schrödinger equation in the critical case

We consider the following quasi-linear Schrödinger equation.

$$\begin{array}{r}i\frac{\mathrm{\partial }\psi }{\mathrm{\partial }t}+\mathrm{△}\psi +\psi \mathrm{△}|\psi {|}^2+|\psi {|}^{p-1}\psi =0,x\in {\mathbb{R}}^D,D\ge 1,(Q)\end{array}$$

where $\psi :{\mathbb{R}}^{+}\times {\mathbb{R}}^D\to \mathbb{C}$ is the wave function, $p=3+\frac{4}{D}$. It is known that the set of standing waves is stable for $1<p<3+\frac{4}{D}$ and it is strongly unstable for $3+\frac{4}{D}<p<\frac{3D+2}{D-2}$. In this paper, we prove that the standing waves are strongly unstable for $p=3+\frac{4}{D}$. Moreover, a property on the set of the ground states of (Q) is investigated.